Show the function T : R² → R³ given by T(x, y) = (x − y, 3x + 2y, 0) is linear.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
1. Show the function \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) given by \( T(x, y) = (x - y, 3x + 2y, 0) \) is linear.
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**Explanation:**
To demonstrate that the function \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) defined by \( T(x, y) = (x - y, 3x + 2y, 0) \) is linear, you need to verify two main properties of linearity:
1. **Additivity:**
\( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
2. **Homogeneity:**
\( T(c\mathbf{u}) = cT(\mathbf{u}) \)
where \( \mathbf{u} \) and \( \mathbf{v} \) are vectors in \( \mathbb{R}^2 \) and \( c \) is a scalar.
Proof of linearity involves the following steps:
### 1. Additivity
Let \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \) be two vectors in \( \mathbb{R}^2 \).
Calculate \( T(\mathbf{u} + \mathbf{v}) \):
\[ \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2) \]
\[ T(\mathbf{u} + \mathbf{v}) = T(u_1 + v_1, u_2 + v_2) \]
\[ = ((u_1 + v_1) - (u_2 + v_2), 3(u_1 + v_1) + 2(u_2 + v_2), 0) \]
\[ = (u_1 - u_2 + v_1 - v_2, 3u_1 + 3v_1 + 2u_2 + 2v_2, 0) \]
Now, calculate \( T(\mathbf{u}) \) and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04829d0-4645-426e-bf1a-7ada40b0786f%2Fbee5c564-981f-4978-bfbb-985b1906084b%2Fhlx7twd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
1. Show the function \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) given by \( T(x, y) = (x - y, 3x + 2y, 0) \) is linear.
---
**Explanation:**
To demonstrate that the function \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) defined by \( T(x, y) = (x - y, 3x + 2y, 0) \) is linear, you need to verify two main properties of linearity:
1. **Additivity:**
\( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
2. **Homogeneity:**
\( T(c\mathbf{u}) = cT(\mathbf{u}) \)
where \( \mathbf{u} \) and \( \mathbf{v} \) are vectors in \( \mathbb{R}^2 \) and \( c \) is a scalar.
Proof of linearity involves the following steps:
### 1. Additivity
Let \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \) be two vectors in \( \mathbb{R}^2 \).
Calculate \( T(\mathbf{u} + \mathbf{v}) \):
\[ \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2) \]
\[ T(\mathbf{u} + \mathbf{v}) = T(u_1 + v_1, u_2 + v_2) \]
\[ = ((u_1 + v_1) - (u_2 + v_2), 3(u_1 + v_1) + 2(u_2 + v_2), 0) \]
\[ = (u_1 - u_2 + v_1 - v_2, 3u_1 + 3v_1 + 2u_2 + 2v_2, 0) \]
Now, calculate \( T(\mathbf{u}) \) and
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